In Exercises 33–34, find and simplify the difference quotient [f(x + h) - f(x)]/h, h =/= 0 for the given function. f(x) = -2x^2 + x + 10

검증된 단계별 안내

Step 1: Start by substituting (x + h) into the function f(x). The function is f(x) = -2x^2 + x + 10, so f(x + h) becomes f(x + h) = -2(x + h)^2 + (x + h) + 10.

Step 2: Expand the expression for f(x + h). Use the distributive property to expand (x + h)^2, which gives x^2 + 2xh + h^2. Substituting this back, f(x + h) = -2(x^2 + 2xh + h^2) + x + h + 10.

Step 3: Simplify f(x + h) by distributing -2 and combining like terms. This results in f(x + h) = -2x^2 - 4xh - 2h^2 + x + h + 10.

Step 4: Compute the difference f(x + h) - f(x). Subtract f(x) = -2x^2 + x + 10 from the expanded f(x + h). This gives [-2x^2 - 4xh - 2h^2 + x + h + 10] - [-2x^2 + x + 10].

Step 5: Simplify the numerator of the difference quotient by canceling out terms and combining like terms. Then divide the simplified numerator by h (noting that h ≠ 0). The final expression will be the simplified form of the difference quotient.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Difference Quotient

The difference quotient is a formula used to calculate the average rate of change of a function over an interval. It is expressed as [f(x + h) - f(x)]/h, where h represents a small change in x. This concept is fundamental in calculus as it leads to the derivative, which measures the instantaneous rate of change.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of the difference quotient, you need to evaluate the function f at both x and x + h. This step is crucial for calculating the difference quotient accurately and understanding how the function behaves as x changes.

Simplification of Expressions

Simplification of expressions is the process of reducing a mathematical expression to its simplest form. This often involves combining like terms, factoring, or canceling common factors. In the context of the difference quotient, simplifying the resulting expression after substituting into the formula is essential for finding a clear and concise representation of the average rate of change.